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I am wondering if the Poincare inequality holds for the Brownian path space.

As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous paths, $w_0 = 0$, , $w_t - w_s \sim N(0, t - s)$ for $t > s$.

Let $E$ be the the collection of continuous functions on $[0, 1]$ with $v(0) = 0$. Of course, $w$ derives a probability measure on $E$, said $\mu$.

Let $f : E \rightarrow \mathbb{R}$ be a function which can be continuously extended to some Frechet differentiable function on $L^2[0, 1]$. By $Df$ I only mean the Frechet derivative.

So my question is that for all $f$ satisfying the condition above, does the following inequality holds with some finite constant $C$: $$\int_E f^2d\mu \leq C\int_E \|Df\|^2d\mu. $$

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    $\begingroup$ In this context I think it's more usual to consider the Malliavin derivative (and its Cameron-Martin norm) instead of Frechét. The Poincaré inequality does hold in that case (of course you have to restrict to $f$ with $\int f\,d\mu = 0$). You can find this, and an involved discussion of the many notions of differentiability (which I have not taken the time to unwind) in Chapter 5 of V. Bogachev's book Gaussian Measures. $\endgroup$
    – Nate Eldredge
    Commented May 1, 2015 at 6:46
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    $\begingroup$ Note also that in the case mentioned by Nate the log-Sobolev inequality holds as well, which gives you quite a bit more than the Poincaré inequality. $\endgroup$
    – Martin Hairer
    Commented May 1, 2015 at 8:24

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